New approach towards different bi-base of ordered b-semiring

The hypothesis of an ordered b-semiring provides a characterization of semiring and ordered semiring in this paper. We have attempted to investigate the Type-1 bi-ideal, Type-2 bi-ideal, Type-1 bi-base and Type-2 bi-base over ordered b-semiring in detail. Some of their characterizations are obtained through Type-1 bi-base and Type-2 bi-base. Let H be a Type-1 (Type-2) hi-base of ordered b-semiring and a, b is an element of H such that a not equal b but not a <= (t1b) b, nor b <= (t1b) a (a <= (t2b) b nor b <= (t2b) a). We introduce the notions of one-quasi order and two-quasi order on an ordered b-semiring to interact with Type-1 and Type-2 bi-bases generated by element and subset. With the help of some examples, we have shown that Type-1 bi-base and Type-2 bi-base have partial order.